Poor Max. Mostly depicted in drawings (though never the star of the show), the Grinch's dog definitely plays second fiddle to his master's nefarious orchestra of doom. The dog follows the Grinch around, never letting the reader know whether or not he is fully on board with the Grinch's plans, though we suspect he is not given his long, morose face.
Max is a very minor character in the grand scheme of things, this is true. Sure, he's nothing more than a pawn in the Grinch's exploits, but he is there to give the reader some pause.
Think about it. Why does the Grinch feel so isolated and excluded, when he has a dog named Max who is clearly loyal enough to go along with his crazy shenanigans?
It goes to show that the Grinch is so intent on his plans to destroy other people's happiness—and so consumed by the idea that he acts alone and has no one—that he fails to recognize the one poor individual who is with him through it all.
Max may be the Grinch's best friend, but the Grinch doesn't seem to recognize it.